What Are Coterminal Angles?

In mathematics, two angles are coterminal, if they have the same terminal and initial sides. The concept of coterminal angles is simple, which will help you to solve many trigonometric problems.

According to the coterminal angle definition, an angle is said to be in standard position if it is drawn on the positive x-axis and turning anti-clockwise. So, the initial side of the angle is a ray where the measurement of an angle starts as well as the terminal side of the angle is a ray where the measurements of the angled end.

Generally, a coterminal angle calculator is drawn at the standard position and shares a terminal side on the graph. For instance, 30 degrees, -330, and 390 degrees are all coterminal.

Coterminal Formula:

The formula to find the Coterminal angles of a given angle is:

Coterminal angles of a given angle θ can be obtained by adding or subtracting a multiple of 360 degrees or 2π radians.

Coterminal of θ in degrees = θ + 360° × k,

Coterminal of θ in radians = θ + 2π × k.

Hence, two angles are coterminal if the difference between angles is a multiple of 2π or 360°. Also, a coterminal angle calculator can obtain the value of a given angle by subtracting or adding a multiple of 360°.

How to find coterminal angles?

We can find the coterminal angles of any angle either by subtracting or adding the multiples of 360 degrees or 2π from the given angle.

We do not need to use the coterminal formula to determine the coterminal angles. Instead, we can either minus or add multiples of 2π or 360 degrees from the given angle to find its coterminal angles.


Determine the coterminal angle of π/4.


The given angle is θ=π/4, which is in the form of radians.

So we either subtract or add multiples of 2π or 360o from it to find its coterminal angles as well as you can find these angles with a coterminal angle calculator.

Let us minus the 2π from the given angle.

π/4 − 2π = −7π/4

Hence, −7π/4 is the coterminal angle of π/4.

Positive and Negative Coterminal Angles:

To find negative and positive coterminal angles, you need to add or minus several complete circles. Then, to find the coterminal angles in the range of 0-360° or 0-2π, just add or subtract 360° or 2π to obtain positive or negative coterminal angles.

For Instance, if θ = 1400°,

then the coterminal angle in the range of 0-360° is 320°.

Which is an example of a positive coterminal angle. So, the other positive coterminal angles are 680°, 1040° and the other negative angles are -40°, -400°. Therefore, you need to simply subtract and add 360° or 2π, if you want any positive and negative coterminal angles.

Why are coterminal angles important?

In mathematics, we often use several functions of angles like sin θ, cos θ, and tan θ. It changes the coterminal angles to have the same values for these trigonometric functions. For instance, 30°, 370°, and -390° are coterminal, and so sin 30°, sin 370° and sin (-390°) and all have the same value when we substitute these angles in the coterminal angle calculator.